Article pubs.acs.org/JPCC
Proton-Conducting Network in Lanthanum Orthophosphate Kazuaki Toyoura,*,† Naoyuki Hatada,† Yoshitaro Nose,† Isao Tanaka,†,‡ Katsuyuki Matsunaga,‡,§ and Tetsuya Uda† †
Department of Materials Science and Engineering, Kyoto University, Kyoto 606-8501, Japan Nanostructure Research Laboratory, Japan Fine Ceramics Center, Nagoya 456-8587, Japan § Department of Materials Science & Engineering, Nagoya University, Nagoya 464-8603, Japan ‡
ABSTRACT: The proton-conducting network in lanthanum orthophosphate, LaPO4, has been theoretically clarified from first principles in the present study. It consists of as many as 20 kinds of migration paths with potential barriers below 1 eV, which are classified into three groups, i.e., rotations and intraand intertetrahedral hoppings. As the results of the kinetic Monte Carlo simulations using the network of the migration paths, the calculated proton diffusion coefficients have anisotropy reflecting the monoclinic crystal structure, particularly large anisotropy in the ca-plane. It results from the dominant proton migration in the directions of [011], [011̅], [111], and [11̅1], whose potential barriers are ∼0.7 eV. The calculated activation energies of the diffusion coefficients in the fastest, the slowest, and the b-axis (unique axis) directions are 0.68, 0.76, and 0.70 eV (= 66, 73, and 68 kJ/mol), respectively, in the temperature range from 500 to 1500 K. These values are lower than the experimentally reported activation energies by conductivity measurements in the range of 0.8−1.0 eV. The major origin of the discrepancy is association effects between protons and dopants. The calculated association energy between protons and strontium dopants is 0.31 eV (30 kJ/mol), which makes the slope of the conductivity curves steeper by 0.21−0.11 eV in the temperature range of 773−1073 K.
I. INTRODUCTION Rare-earth phosphates have been reported as a class of proton conductors,1−7 which are candidate materials for solid electrolytes in fuel cells at intermediate temperatures (473−973 K). Lanthanum orthophosphate, LaPO4, is one of the protonconducting phosphates, exhibiting predominantly protonic conductivity up to 973 K in wet air by introducing negatively charged defects, e.g., substitution of lower-valent cations at La3+ sites.8 Actually, alkaline-earth-metal-doped LaPO4 has been reported to exhibit high conductivity close to 10−4 S/cm at 973 K.1,2,9 In addition, it has a great advantage in thermodynamical stability over other proton-conducting lanthanum phosphates, such as polyphosphate and ultraphosphate (LaP3O9 and LaP5O14) with gradual pyrolyses at intermediate temperatures.10,11 LaPO4 is, therefore, expected as a promising candidate for solid electrolytes in intermediate-temperature fuel cells. As for proton incorporation into LaPO4, defect equilibria including dopants of alkaline earth metals have been proposed as the following reactions12 1/2M 2P2O7 → M′La + 1/2(P2O7 )•• 2(PO4 )
(1)
• 1/2(P2O7 )•• 2(PO4 ) + 1/2H 2O → (HPO4 )PO4
(2)
dopant dissolution according to the charge compensation mechanism. The existence of P2O7 units was experimentally observed in Sr-doped LaPO4 by Fourier transform Raman (FTRaman) spectroscopy and magic-angle spinning nuclear magnetic resonance (MAS-NMR) spectroscopy measurements. 12 The following reaction is hydration at the pyrophosphate ions (P2O74−), leading to formation of hydrogen diphosphate ions (HPO42−). The expression of HPO42− means that protons reside around PO4 units in LaPO4. The proton transfer mechanism in LaPO4 was theoretically addressed by Yu and De Jonghe using first-principles calculations.14 They classified the proton transfer paths into four types, i.e., rotational, oscillatory, intratetrahedral, and intertetrahedral proton transfers, and concluded that the intertetrahedral proton transfer paths with 0.6−1.0 eV potential barriers play a key role in the proton conduction. Furthermore, the diffusion coefficient and conductivity of protons was estimated using assumptions of isotropic random walks, the average potential barrier, 0.8 eV, and negligible interactions between protons and dopants. Their computational results roughly agree with the experimental reports by conductivity measurements, 0.8−1.0 eV.1,8,9,12,15,16
where M denotes an alkaline earth metal and the defects are described according to Kröger−Vink notation.13 In this model, the first reaction is condensation of orthophosphate ions by © 2012 American Chemical Society
Received: May 21, 2012 Revised: August 15, 2012 Published: August 20, 2012 19117
dx.doi.org/10.1021/jp304904j | J. Phys. Chem. C 2012, 116, 19117−19124
The Journal of Physical Chemistry C
Article
Figure 1. (a) Crystal structure of LaPO4 with the isosurface in the calculated potential energy surface (PES) of protons and the energy local minima, i.e., proton sites. The green balls and the purple tetrahedra denote La ions and PO4 units, respectively. The oxygen sites at the corners of PO4 are classified into four, O1, O2, O3, and O4, shown by blue, pink, black, and red balls, respectively. The isosurface level of the yellow surface is 1.3 eV with reference to the most stable. The small balls bonding to oxide ions are the proton sites. (b) The proton sites around each oxide ion. The energies in the parentheses are the potential energies with reference to the most stable.
corresponding potential barriers. These mean jump frequencies, ν, were estimated using the following equation on the basis of classical mechanics
In the present study, we theoretically investigate the proton conduction mechanism in LaPO4 in a more rigorous manner from first principles. Possible proton migration paths in the crystal are totally taken into consideration, though such an exhaustive survey has been avoided so far in complicated crystal structures with low symmetry. Furthermore, proton diffusion simulations are performed using the migration paths by means of the kinetic Monte Carlo methods,17,18 to estimate the diffusion coefficients and conductivities of protons as a function of temperature. Through these theoretical analyses, we have found several key migration paths determining the proton mobility and leading to the anisotropic diffusion coefficients and conductivities in LaPO4. In addition, association effects between protons and dopants on the concentration of mobile protons are discussed, to compare our computational results with the previous experimental reports.
⎛ ΔEmig ⎞ ν = ν* exp⎜ − ⎟ ⎝ kBT ⎠
(3)
where kB is the Boltzmann constant; T is the temperature; and ΔEmig is the potential barrier of the proton jumps. ν* is the vibrational prefactor, which is related to the lattice vibrations at the initial and saddle-point states on the basis of the transition state theory (TST).27−30 In the present study, the vibrational prefactors of the proton migration paths were assumed to be 1013 THz according to the theoretical report on the vibrational prefactors in LaPO4.14 The proton diffusion coefficients were estimated by the following definitional equation for independent particles
II. COMPUTATIONAL PROCEDURES All the computational studies were based on first-principles calculations using the projector-augmented wave (PAW) method19 implemented in the VASP code.20−24 The generalized gradient approximation (GGA) parametrized by Perdew, Burke, and Ernzerhof was used for the exchange-correlation term.25 The plane wave cutoff energy was 400 eV. The 5s, 5p, 6s, and 5d orbitals for lanthanum, 4s, 4p, and 5s for strontium, 3s and 3p for P, 2s and 2p for oxygen, and 1s for hydrogen were treated as valence states. A supercell consisting of 2 × 2 × 2 unit cells of LaPO4 was used with a single k-point sampling at Γ-point. For investigation of the proton conduction in LaPO4, first of all, it is necessary to explore proton sites, i.e., energy local minima of protons, in the crystal. They were determined by construction of the potential energy surface (PES) of protons with the fixed atomic positions and the subsequent structural optimization. A 20 × 20 × 20 grid per unit cell was used for the PES construction. The energy local minima in the PES were used as the initial structures in the subsequent structural optimizations. Atom positions were fully optimized until the residual forces became less than 0.02 eV/Å. The proton migration paths and their energy profiles were evaluated using the nudged elastic band (NEB) method,26 taking all the possible paths connecting the proton sites within 3.5 Å distance into consideration. The kinetic Monte Carlo (KMC) simulations17,18 were finally performed using the mean jump frequencies of the migration paths estimated from the
Dxi = lim
n →∞
rx2i , n 2tn
(4)
where xi denotes a direction in the crystal and tn and rxi,n are the time and the xi-component of the displacement vector after ntimes jumps. The KMC steps, n, were carefully determined in the range from 8 × 103 to 1 × 108 steps by checking convergence of the diffusion coefficient at each temperature. The association energy between protons and strontium dopants was evaluated from the energy difference between the two supercells including both a proton and a strontium ion with the nearest and furthest H+−Sr2+ distances. The relative energies of proton sites in the furthest region from the strontium ion coincide with those in the undoped supercell within 0.01 eV differences, which indicates the supercell size employed in the present study is sufficient for estimation of the association energy.
III. RESULTS A. Energy Local Minima of Protons. The crystal structure of LaPO4 (Figure 1a) is a monoclinic monazite structure (space group: P21/n). The calculated lattice parameters, a, b, c, and β, are 6.93 Å, 7.15 Å, 6.54 Å, and 103.67°, which coincide with the experimental values31 within 1.5%. Phosphorus and oxygen atoms make strong covalent bonds, to form PO4 tetrahedral units. All the PO4 tetrahedra isolated in the crystal are crystallographically equivalent. Four oxygen sites located at the 19118
dx.doi.org/10.1021/jp304904j | J. Phys. Chem. C 2012, 116, 19117−19124
The Journal of Physical Chemistry C
Article
Table 1. Calculated Migration Paths with Low Potential Barriers below 1 eV with Reference to the Potential Energy at the Most Stable Site, O4-3 path
a
potential barrier height (eV)
no.
initial site
final site
distance (A)
classification
vs O4-3
vs initial site
vs final site
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
O4-3 O1-2 O4-3 O4-2 O3-1 O4-1 O3-3 O4-3 O1-2 O3-2 O1-1 O1-2 O4-1 O4-3 O4-1 O4-1 O4-1 O4-3 O2-1 O2-1
O1-2 O1-2 O2-1 O3-1 O3-2 O4-2 O1-1 O4-2 O3-1 O3-3 O1-1 O4-1 O1-1 O2-2 O3-3 O2-1 O3-1 O4-1 O2-2 O3-2
0.60 1.64 1.47 0.46 0.84 1.50 0.41 1.81 2.35 1.44 1.96 2.99 2.78 1.89 2.19 2.15 2.57 2.35 2.25 2.82
tntertetrahedral hoppinga intertetrahedral hopping intertetrahedral hopping intertetrahedral hoppinga rotation rotation intertetrahedral hoppinga rotation intertetrahedral hopping rotation intertetrahedral hopping intertetrahedral hopping intertetrahedral hopping intertetrahedral hopping intertetrahedral hopping intertetrahedral hopping intertetrahedral hopping rotation rotation intertetrahedral hopping
0.09 0.26 0.44 0.44 0.49 0.53 0.54 0.56 0.64 0.67 0.69 0.70 0.79 0.83 0.87 0.88 0.90 0.92 0.93 0.97
0.09 0.19 0.44 0.02 0.06 0.38 0.03 0.56 0.57 0.23 0.16 0.63 0.64 0.83 0.72 0.73 0.75 0.92 0.62 0.66
0.02 0.19 0.13 0.01 0.05 0.11
